Dear Gil,
In my opinion, you should use the correlated posterior marginals since they should produce the more valid results.
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Even if initially the Young’s moduli E_1 and E_2 are really not correlated: Since you have the observation of a a settlement of 1.5 cm you should use for further computation samples (E_{1,i,}E_{2,i}) for (E_1,E_2) that produce approximation of this settlement, i.e. those pairs that “match” the observation of 1.5 cm. This is similar to the situation in Bayesian inversion : limit the correlation of posterior inputs.
And to get sample pairs of this kind, you need that the random variables representing E_1 and E_2 are correlated.
If you only use the uncorrelated posterior marginals, and use then the corresponding mean values to computed the settlement you may end up in s situation similar to the one in Example of Bayesian inference computation creating a model output at the mean of the parameter samples that is outside of the discrete posterior predictive support. -
Minor remark: I am not sure that one can really rule out thatt the Young’s moduli E_1 and E_2 are correlated for physical reasons: The two soil layers are connected at some interface such that one may have a material exchange between the layers through this interface changing the material properties of the soil near to the interface such that the effective Young’s moduli you seem to use in your computation may become correlated.
Greetings
Olaf